Option #3 (quit and work on projects) also seems to come with a salary cut (down to 0). If working for a startup would drastically cut your family's standard of living, so would this option.
Startups have other problems besides the lower salaries you were offered: the vast majority of all startups fail quickly, so there's no job security. If you take one of these jobs, you might be looking for a new job in a few months.
I'd suggest staying at your current job and continuing to look for another one, but not at a startup.
Also, you say that your current company isn't working out for you. Is it really the company, or just the job you have there? Would it be possible to change jobs within your current company?
I've racked my brain for months before starting my job search, even talking to coworkers and my manager. Basically it's a cultural issue, and I dont mean just company culture, but Korean culture. Not being Korean, I don't speak Korean fluently and I don't know the cultural norms. This means that I am unable to form any friendships or close relationships with the people I work with. Korean culture is heavily driven by relationships, and not being able to form them means it's hard to get things done. So it ends up that people route their jobs around me, talking to my direct reports or superiors instead of talking to me. Also, responses to my requests are slow or nonexistent, and I have to route my own requests around me also. These and other issues relegate me to the role of researcher and consultant, where I am on the periphery of the job I am supposed to be doing. If I had less ambition I might be perfectly happy to just coast here and collect the paycheck, but the lack of working towards something worthwhile is frankly giving me a mild case of depression.
Startups have other problems besides the lower salaries you were offered: the vast majority of all startups fail quickly, so there's no job security. If you take one of these jobs, you might be looking for a new job in a few months.
I'd suggest staying at your current job and continuing to look for another one, but not at a startup.
Also, you say that your current company isn't working out for you. Is it really the company, or just the job you have there? Would it be possible to change jobs within your current company?